The amount of scattering and absorption by a particle is usually expressed in
terms of the scattering cross section Cs and absorption cross section Ca. The total
amount of absorption and scattering, or extinction, is expressed in term of the
extinction cross section Ct. The dimensionless efficiency factors are often used
instead of cross sections,

(1)

Absorption and scattering of radiation by a two-layer spherical particle depend on
the complex index of refraction of the core and mantle substances (m' = n'-iκ' and
m'' = n''- iκ'', correspondingly) and also on diffraction parameters x' = 2πa'/λ,
x'' = 2πa''/λ, where a' is the core radius, and a'' is the external radius of the
particle.

According to Mie theory, the efficiency factors of scattering and extinction are
expressed in the form of the following series:

(2)

where complex coefficients ak, bk are called the Mie coefficients. The efficiency factor
of absorption is determined as a difference, Qa = Qt - Qs. The terms in the Mie
series correspond to the partial waves of different orders. The number of terms, which
should be taken into account, increases with increasing the diffraction parameter. As
a result, the calculations for particles of radius much greater than the wavelength are
more complicated.

The angular characteristics of scattering are expressed by the following
complex amplitude functions corresponding to perpendicular polarizations:

(3a)

(3b)

Here, μ = cosθ, where θ is the angle of scattering measured from the direction of the
incident radiation (see Fig. 1), and πk and τk are special angular functions defined
later by Eq. (22). The scattering (phase) function for linear polarized incident
radiation is

(4)

where

(5)

Angle φ is measured from the polarization plane of the incident radiation. For
unpolarized (randomly polarized) incident radiation, the following relations hold
true:

(6)

and the polarization degree of scattered radiation is

(7)

For calculating the asymmetry factor of scattering, one can use the Debye
equation,

(8)

where the asterisk denotes a complex conjugate quantity. Remember that the
asymmetry factor of scattering is defined as

(9)

According to Eq. (8), the value of μ does not depend on polarization of the incident
radiation.

The Mie coefficients for two-layer spherical particles are determined by

(10)

where

(11)

and αk, βk, γk expressions will be given later [see Eqs. (17) and (18)].

For hollow particles, Eqs. (10) and (11) become

(12)

(13a)

(13b)

(13c)

(13d)

For homogeneous particles, the simplifications of the Mie coefficients are
considerable,

and can be calculated following the recursion relations derived by Hosemann
(1971),

(23)

For calculations employing Eqs. (19) and (23), it is necessary to know the following
initial functions:

(24a)

(24b)

(24c)

We have now all the expressions for calculations of absorption and scattering of
radiation by homogeneous, hollow, or two-layer spherical particles.

The calculations of the Riccati-Bessel functions based on recursion relations (19)
starting from the first functions [(24a) and (24b)] up to higher-order functions (the
so-called upward recursion) may lead to significant computational errors. The latter limitation is
important in the case of large values of x and |m|x when the functions are of the
order close to the argument are calculated with great error. This problem was
overcome by Kattawar and Plass (1967), who showed that some components of the
Riccati-Bessel functions are well calculated by using downward recursion. The
detailed analysis of the accuracy and stability of several algorithms for Mie scattering
calculations have been performed by Wiscombe (1980). A comparison of Mie
scattering subroutines can be found in the paper by Felske et al. (1983). The reader
can find the early FORTRAN codes for homogeneous and various two-layer particles
in appendices of the books by Bohren and Huffman (1983) and Dombrovsky
(1996). The algorithms for the general case of stratified spheres were presented
by Toon and Ackerman (1981) and Bhandari (1985). Additional codes for
multilayered spheres are listed by Flatau (2000) and Wriedt (2000). Note
that the work on improving the Mie scattering algorithms continuesd. One
can find some new results in recent papers by Yang (2003), Du (2004), Li
et al. (2006), and Cai et al. (2008). A more detailed bibliography on this
subject can be found in the recent monograph by Dombrovsky and Baillis
(2010).

Fortunately, this advanced technique of the Mie calculations should only be used
mainly in the limiting cases when the general solution is degenerated and reliable
estimates can be obtained on the basis of known approximations. The geometrical
optics approximation for very large particles is a good illustration of the latter
statement, which is a particular case of the general observation, namely, one should
find an alternative physical approach in the range when the ordinary procedure leads
to more and more complicated mathematics.